Physics · Kinetic Theory · NEET
A monatomic atom is just one point mass. It can only move (translate) in 3 directions: x, y and z. So it has 3 degrees of freedom. By the law of equipartition, each degree of freedom carries 1/2 kB T of energy per molecule, or 1/2 RT per mole. So internal energy U = 3 x (1/2 RT) = 3/2 RT per mole. Since Cv = dU/dT, we get Cv = 3/2 R. There is no rotation or vibration to add, because a single point atom has nothing to spin or stretch.
Use Mayer's relation for an ideal gas: Cp - Cv = R. So Cp = Cv + R = 3/2 R + R = 5/2 R. In plain numbers, Cv = 12.47 J per mol per K and Cp = 20.79 J per mol per K, using R = 8.314. Cp is always larger than Cv because at constant pressure the gas also does work by expanding, so extra heat is needed for the same rise in temperature.
Gamma is just the ratio of the two specific heats. gamma = Cp/Cv = (5/2 R) / (3/2 R) = 5/3 = 1.67. Notice R cancels out, so gamma does not depend on which monatomic gas it is. Helium, Argon, Neon, Krypton all have gamma = 5/3 as long as they behave ideally. A quick shortcut: gamma = 1 + 2/f, where f is degrees of freedom. For monatomic f = 3, so gamma = 1 + 2/3 = 5/3.
No. Molar specific heats Cv = 3/2 R and Cp = 5/2 R are the same for He, Ar, Ne and any ideal monatomic gas, because they depend only on degrees of freedom, not on mass. Heavier atoms move slower but carry the same energy per degree of freedom at the same temperature. Note: the SPECIFIC heat per kilogram does differ between gases, because it divides the molar value by molar mass. NEET usually asks for molar specific heat, so use 3/2 R and 5/2 R.
Rotation needs a body with size that can spin about an axis and store rotational energy. A single atom is treated as a point mass, so spinning it stores no meaningful energy (its moment of inertia about its own centre is taken as zero). Vibration needs two or more atoms joined by a bond that can stretch. A monatomic gas has no bond. So only the 3 translational modes are active, giving f = 3.
The average thermal energy for a mono-atomic gas is: (kB is the Boltzmann constant and T the absolute temperature)
Try the real previous-year questions from this chapter — each with the answer and a full solution.
For helium, a monatomic ideal gas, the molar specific heat at constant volume is Cv = 3/2 R = 12.47 J per mol per K (using R = 8.314). The same value holds for argon, neon and other monatomic gases.
Cp = 5/2 R = 20.79 J per mol per K. It comes from Cp = Cv + R = 3/2 R + R.
Argon is monatomic, so gamma = Cp/Cv = 5/3 = about 1.67, the same as helium and neon.
Smaller. Monatomic Cv = 3/2 R while diatomic Cv = 5/2 R, because diatomic molecules have 2 extra rotational degrees of freedom that also store energy.
Three, all translational (motion along x, y and z). There is no rotation or vibration for a single point atom, so f = 3.
